If $A, B, C$  be the angles of a triangle, then $\left| {\,\begin{array}{*{20}{c}}{ – 1}&{\cos C}&{\cos B}\\{\cos C}&{ – 1}&{\cos A}\\{\cos B}&{\cos A}&{ – 1}\end{array}\,} \right| = $

Let $A = \left[ {\begin{array}{*{20}{c}}
  2&b&1 \\ 
  b&{{b^2} + 1}&b \\ 
  1&b&2 
\end{array}} \right]$  where $b > 0$. Then the minimum value of $\frac{{\det \left( A \right)}}{b}$ is

Find area of the triangle with vertices at the point given in each of the following: $(-2,-3),(3,2),(-1,-8)$

The system of equations  $-k x+3 y-14 z=25$  $-15 x+4 y-k z=3$  $-4 x+y+3 z=4$  is consistent for all $k$ in the set

Let $D _{ k }=\left|\begin{array}{ccc}1 & 2 k & 2 k -1 \\ n & n ^2+ n +2 & n ^2 \\ n & n ^2+ n & n ^2+ n +2\end{array}\right|$. If $\sum \limits_{ k =1}^n$ $D _{ k }=96$, then $n$ is equal to